They (hill climb teams) are the absolute experts at harrying a car up the hill, be it an F2 car, caterfield type or standard saloon style; they'd show the F1 drivers a clean set of heels.
The hill climb people are experts at just that, as you have stated. PaulR on here did this for a few years and had a very fast car. A friend of mine at Sauber used to do this as well. He came second in the UK championship quite a few years ago and then made a bad choice in terms of an engine to try and win the following year.
The Pikes peak video is great (thanks for posting), but the car looks like it would have gone even faster with a better setup.
As to racing an F1 car up such peaks, they are not designed to do that at all. However, if that was set as an objective, then a modified F1 car would be extremely fast as well. It would just be a question of the rules, the budget, the resource and the time available.